Grand Canonical Versus Canonical Ensemble: Universal Structure of Statistics and Thermodynamics in a Critical Region of Bose–Einstein Condensation of an Ideal Gas in Arbitrary Trap

Abstract

We find a self-similar analytical solution for the grand-canonical-ensemble (GCE) statistics and thermodynamics in the critical region of Bose–Einstein condensation. It is valid for an arbitrary trap, loaded with an ideal gas, in the thermodynamic limit. We show that for the quantities, changing by a finite amount across the critical region, the exact GCE result differs from the corresponding canonical-ensemble result by a factor on the order of unity even in the thermodynamic limit. Thus, a widely used GCE approach does not describe correctly the critical phenomena at the phase transition for the actual systems with a fixed number of particles and yields only an asymptotics far outside the critical region.

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Acknowledgments

A support from RFBR (Grant 12-02-00855-a), Program of fundamental research of the Physical Science Branch of the Russian Academy of Science (section III.7) and the Council on grants of the President of the Russian Federation for support of the leading scientifc schools of the Russian Federation (Grant HIII-1041.2014.2) is acknowledged.

Appendix: GCE and CE Asymptotics for a Mesoscopic System Far Outside the Critical Region in the Disordered Phase

We show that, in the disordered, classical phase (\(T>T_c\)) in a large enough mesoscopic system, any thermodynamic function has the same GCE and CE asymptotics at large negative values of the self-similar variable (12). We follow the saddle-point method, described in [99] and [51], but show this fact for a mesoscopic system allowing the limit \(-\eta \gg 1\) of smaller than critical number of particles or larger than critical temperature, instead of taking the thermodynamic limit \(\alpha \rightarrow 0\). We consider the average energy in the leading order. A generalization to other thermodynamic functions is straightforward.

First, it is easy to check that the stated asymptotics for the CE average energy \(E^{(ce)}=T^2 \partial \ln Z /\partial T\), written via the CE partition function (10), \(Z=Z^{(\infty )}P^{(\infty )}(N)\), is given by a following formula

Far from the critical region into the disordered phase at \(-\eta \gg 1\), it has a pure imagine solution \(u_{st}= i \ \text {Im}u_{st}\) with a very large positive imagine part \(\text {Im }u_{st} \sim N_c /\sigma \gg 1\). That allows one to neglect an exponentially small value \(\exp (iu_{st}(N+1))\) in Eq. (44) and rewrite it in the form

Equation (47) can be used even for a mesoscopic system with finite \(\alpha \). In the thermodynamic limit \(\alpha \rightarrow 0\), it reveals the chemical-potential critical function:

$$\begin{aligned} i u_{st} = F_{\mu }(\eta ). \end{aligned}$$

(48)

Finally, plugging the result (47) into Eq. (41), we conclude that, indeed, at \(-\eta \gg 1\) the CE average energy asymptotically tends to the GCE average energy in Eq. (35) for large enough mesoscopic system. That GCE-CE asymptotic equivalence in the disordered phase at \(\eta \rightarrow -\infty \) takes place also for the heat capacity and other thermodynamic quantities whenever the saddle-point method for calculation of \(P^{(\infty )}\) is valid. It also holds for the universal critical functions of thermodynamic quantities, defined as in Eq. (34). The latter is obvious from Eq. (40) and the following universal relation between the scaled chemical potential \(F_{\mu }\) and the universal distribution \(P_{\eta }^{(\infty )}\) (Eq. (29)),

Napolitano, R., De Luca, J., Bagnato, V.S., Marques, G.C.: Effect of a finite number of particles in the Bose–Einstein condensation of a trapped gas. Phys. Rev. A 55, 3954–3956 (1997)CrossRefADSGoogle Scholar